definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its double dual space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its double dual space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( V\): \(\in \{\text{ the finite-dimensional } F \text{ vectors spaces }\}\)
\( V^*\): \(= L (V: F)\)
\( {V^*}^*\): \(= L (L (V: F): F)\)
\( J\): \(\in \{\text{ the finite index sets }\}\)
\( B\): \(\in \{\text{ the bases for } V\}\), \(= \{b_j \vert j \in J\}\)
\( B^*\): \(= \text{ the dual basis of } B\), \(= \{b^j \vert j \in J\}\)
\( {B^*}^*\): \(= \text{ the dual basis of } B^*\), \(= \{\widetilde{b}_j \vert j \in J\}\)
\(*f\): \(: V \to {V^*}^*, v^j b_j \mapsto \sum_{j \in J} v^j \widetilde{b}_j\), \(\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}\)
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Conditions:
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\(f\) does not depend on the choice of \(B\): compare to the definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its covectors space with respect to original space basis with "with respect to original space basis".
2: Note
\(f\) is indeed a 'vectors spaces - linear morphisms' isomorphism, does not depend on the choice of \(B\), and has the property that for each \(v \in V\) and each \(w \in V^*\), \(w (v) = f (v) (w)\), by the proposition that between any finite-dimensional vectors space and its double dual, there is the canonical 'vectors spaces - linear morphisms' isomorphism.
Although it is sometimes sloppily expressed like "The double dual of a finite-dimensional vectors space is the original vectors space.", the double dual is not the same entity with the original vectors space, as the 2 have different meanings. They are just 'vectors spaces - linear morphisms' isomorphic, and having such a relation does not make 2 entities the same.
